Televiewer image wood-grain reduction techniques

ABSTRACT

Measurements made by a transducer assembly for downhole imaging are affected by reverberations between the transducer and the window on the outside of the assembly. The reverberations result in a stationary noise on the image. Hardware solutions to improve signal-to-noise ratio includes using a composite transducer, adjusting the distance between the transducer and the window. SNR can also be improved by processing techniques that include stacking, fitting a sinusoid to the reverberation, by envelope peak detection, and by applying a notch filter.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of application Ser. No. 12/760,333filed Apr. 14, 2010 which in turn claims priority from U.S. ProvisionalPatent Application Ser. No. 61/171,220 filed on Apr. 21, 2009, both ofwhich are incorporated herein by reference in their entirety.

FIELD OF THE PRESENT DISCLOSURE

A downhole acoustic logging tool is provided for imaging the texture andstructure of the borehole sidewall. The signal components due totransducer ringdown and internal tool reverberations are processed toprovide an image in which the “wood-grain” image artifact is reduced.

BACKGROUND OF THE PRESENT DISCLOSURE

Typical acoustic logging tools may include, by way of example, ateleviewer which comprises a rotating ultrasonic acoustic transducerthat operates in a frequency range on the order of 100 kHz or more.Higher acoustic frequencies are preferred in order to achieve betterresolution in the confined space of a borehole. In operation, theteleviewer rotates at a desired rate such as 5 to 16 rotations persecond to continuously scan the borehole sidewall as the televiewer isdrawn up the borehole at a rate that is typically 3/16 to ⅜ inch perscan. A beam of acoustic pulses is launched along the normal to theborehole sidewall as the transducer scans the interior surface of theborehole. The pulse rate depends upon the desired spatial resolutionsuch as 1500 pulses per second or 128 to 256 pulses per scan. Theinsonified borehole sidewall returns pulses reflected therefrom, back tothe transducer on a time-multiplexed basis. The reflected acousticsignals are detected, amplified and displayed to provide a continuouspicture of the texture and structure of the borehole sidewall. Otherapplication include determination of the goodness of a cement bond to asteel casing as well as monitoring the integrity of the casing itself.

The diameter of a borehole logger is on the order of 2% in (7.3 cm), sothat it can be run into relatively small boreholes. However manyborehole diameters are on the order of 10-14″ (25.4-35.6 cm) or more sothat the length of the acoustic-pulse trajectory from the transducer,through the borehole fluid to the borehole sidewall, may be up to 10″(25.4 cm). In the normal course of events, the borehole fluid iscontaminated by drill cuttings, air bubbles and foreign matter whichseverely attenuate the acoustic energy by scattering because thephysical dimensions of the contaminants are comparable to the wavelengthof the wavefields emitted by the transducer.

The televiewer signal is also contaminated by stationary noise. Thestationary noise is due to the transducer ringing and the reverberationsbetween the face of the transducer and a window on the outside of theteleviewer assembly. The echo signal is the signal returned from theborehole formation or casing. Neglecting borehole fluid attenuation,deflection, diffraction, and window insertion loss, the amplitude of theecho signal is a function of the acoustic impedance of the formation orcasing. The arrival time of the echo signal within the process windowwill change with the instruments centralization and the shape of theborehole. As the echo signal moves in time, it is modulated by thestationary noise, therefore the detected peak amplitude is a function ofthe formation reflection, and the position of the echo within theprocess window. If the S/N ratio is low enough, the modulation causesthe resulting televiewer amplitude image to be dominated by the periodic“wood grain” interference pattern. A physical explanation of the “woodgrain” is that the transducer ringing and reverberation noise isinitiated when a pulse-echo signal is transmitted, so that the noise isstationary with respect to the echo signal. Strictly speaking, the term“ringdown” refers to the ringing of the transducer when it is activated.Reverberation refers to the sound that reflects back and forth betweenthe transducer and window. Interference between the ringdown andreverberation noise and the echo signals results in a “wood grain”pattern on the amplitude image. The present disclosure addresses theproblem of reducing the “wood grain” effects by improvements to theteleviewer assembly, detecting the signal independently of the noise inreal time, and/or filtering the stationary noise post acquisition.

SUMMARY OF THE PRESENT DISCLOSURE

One embodiment of the disclosure is an apparatus configured to evaluatean earth formation. The apparatus includes a rotatable transducerassembly; a transducer on the rotatable transducer assembly configuredto propagate an acoustic signal through an acoustically transparentwindow into a borehole and produce a signal including a echo from a wallof the borehole and a ringdown signal within the transducer assembly;and at least one processor configured to use the signal produce at aplurality of orientations of the transducer during rotation of thetransducer assembly to provide a two-dimensional image of the earthformation, the two-dimensional image being substantially free of astationary noise resulting from an interference between the ringdownsignal and the echo.

Another embodiment of the disclosure is a method of evaluating an earthformation. The method includes: conveying a rotatable transducerassembly into a borehole; using a transducer on the rotatable transducerassembly to convey an acoustic signal through an acousticallytransparent window into a borehole and produce a signal including enecho from a wall of the borehole and a ringdown signal within thetransducer assembly; and processing the signal produced at a pluralityof orientations of the transducer during rotation of the transducerassembly to provide a two-dimensional image of the earth formation, thetwo-dimensional image being substantially free of a stationary noiseresulting from interference between the ringdown signal and the echo.

Another embodiment of the disclosure is a computer-readable mediumproduct having stored thereon instructions that when executed by atleast one processor, cause the at least one processor to perform amethod. The method includes: using a signal produced by a transducer ona rotating transducer assembly in a borehole, the signal including anecho from a wall of the borehole and a ringdown signal within therotating transducer assembly at a plurality of orientations of thetransducer assembly, for providing a two-dimensional image of the earthformation, the two-dimensional image being substantially free of astationary noise resulting from the ringdown signal.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure is best understood with reference to theaccompanying figures in which like numerals refer to like elements andin which:

FIG. 1 shows an imaging well logging instrument disposed in a wellboredrilled through earth formations.

FIG. 2A shows the rotator assembly;

FIG. 2B shows the transducer assembly;

FIG. 3 illustrates a cross-section of the transducer assembly;

FIG. 4A shows an exemplary signal resulting from activation of thetransducer;

FIG. 4B shows an exemplary amplitude image of the borehole walldominated by an interference between the transducer ringdown and theecho signal, giving a characteristic “wood-grain” image;

FIG. 5A shows the amplitude and traveltime images obtained using aceramic transducer in heavy water-based mud;

FIG. 5B shows the amplitude and traveltime images obtained using acomposite transducer in heavy water-based mud;

FIG. 6A shows an exemplary signal including a ringdown and an echosignal;

FIG. 6B is an estimate of the ringdown obtained by fitting a sinusoid;

FIG. 6C shows the estimated echo signal obtained by subtracting thefitted sinusoid from the recorded signal;

FIGS. 7A-D shows four exemplary recorded signals, each including an echoand a ringdown,

FIGS. 7E-H shows the results of subtracting the average of the signalsin FIGS. 7 a-d from the individual signals;

FIGS. 8A-8C show the results of processing a signal including a ringdownusing ABS, Stack and subtract, and Fit and subtract respectively;

FIGS. 9A-9C show exemplary signals including a ringdown and an echo atdifferent spacings from the ringdown;

FIG. 10 shows the Cauchy wavelet pair for the Hilbert Transform

FIG. 11A shows an exemplary signal dominated by a ringdown;

FIG. 11B shows the envelope of the signal in FIG. 10 a;

FIG. 12 shows the envelope along with the result of applying a GaussianLaplace operator;

FIG. 13 shows the geometry of an eccentric rotating transducer in acircular borehole;

FIG. 14 shows the effect of eccentering of the transducer in a circularborehole; (a) shows the peak amplitude as a function of azimuth as aresult of eccentering of the transducer by 24 mm in a borehole ofdiameter 108 mm, (b) shows the woodgrain effect, and (c) shows thespectrum of the woodgrain effect;

FIGS. 15A-15B show a comparison of amplitude images obtained withoutdownhole waveform processing and with stack-and-subtract waveformprocessing.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

Referring to FIG. 1, a well logging instrument 10 is shown being loweredinto a wellbore 2 penetrating earth formations 13. The instrument 10 canbe lowered into the wellbore 2 and withdrawn therefrom by an armoredelectrical cable 14. The cable 14 can be spooled by a winch 7 or similardevice known in the art. The cable 14 is electrically connected to asurface recording system 8 of a type known in the art which can includea signal decoding and interpretation unit 16 and a recording unit 12.Signals transmitted by the logging instrument 10 along the cable 14 canbe decoded, interpreted, recorded and processed by the respective unitsin the surface system 8.

FIG. 2A shows mandrel section 201 of the imager instrument with aTeflon® window 203. Shown in FIG. 2B is the rotating platform 205 withthe ultrasonic transducer assembly 209. This may be referred to as arotatable transducer assembly. The rotating platform is also providedwith a magnetometer 211 to make measurements of the orientation of theplatform and the ultrasonic transducer. The platform is provided withcoils 207 that are the secondary coils of a transformer that are usedfor communicating signals from the transducer and the magnetometer tothe non-rotating part of the tool. The transducer 209 is discussedfurther below.

Turning now to FIG. 3, a cross section of a transducer assembly isshown. Depicted therein is the Teflon® window 301. One transducer isdepicted by 307. The assembly may include a second transducer on theopposite side (not shown). The front portion of the transducer is incontact with an impedance matching material 305 that is used to matchthe impedance of the transducer with that of oil in the space 305between the transducer and the Teflon® window. It should be noted thatthe use of Teflon® is not to be construed as a limitation and any othermaterial with the necessary abrasion resistance and acoustic propertiedcould be used.

Still referring to FIG. 3, the transducer assembly also includes abacking material 309. In one embodiment, the backing material is a 0-3composite of tungsten particles in high temperature rubber. In anotherembodiment, liquid Viton®, a synthetic rubber may be used. The backingmaterial absorbs acoustic signals propagating from the transducer awayfrom the borehole wall and reduces reflections from the interfacebetween the transducer and the backing material. Also shown in FIG. 3are the leads 311 from the transducer that go to transformer coils 207.The transducer 307 sends an acoustic signal through the acousticallytransparent Teflon® window. The borehole wall produces an echo signalthat returns to the rotatable transducer assembly 209.

A problem encountered with televiewer signals is that of reverberationswithin the transducer assembly, specifically between the transducer 307and the Teflon® window 301. FIG. 4A shows an exemplary signal recordedwith a rotatable transducer assembly 209. The reflection from theborehole wall is indicated by 401 while the extended signal 403 resultsfrom ringdown of the transducer. As a result of interference betweenthis ringdown and the echo signal, the amplitude image of the boreholewall has the character shown in FIG. 4B. The abscissa of thetwo-dimensional image of FIG. 4 b is depth and the ordinate is theazimuth: the borehole wall has been “unwrapped” to give the image. Theactual reflection signals from the borehole wall corresponding to twodipping beds or fractures are shown by 411 and 413. The image isdominated by a “wood-grain” pattern 409 which makes the identificationof the dipping beds difficult.

In the present disclosure, several methods are disclosed for improvingthe signal-to-noise ratio of the desired signal (exemplified by 411) andthe noise (exemplified by 409). The first solution is a hardwaresolution and is based upon using a transducer that has improved signaloutput. Specifically, a composite transducer of the type disclosed inU.S. patent application Ser. No. 12/392,487 of Steinsiek et al., havingthe same assignee as the present disclosure and the contents of whichare incorporated herein by reference, is used. The transducer disclosedtherein comprises a 1-3 Piezoelectric composite transducer of high Qceramic rods in a polymer matrix. A comparison of signals recorded in aborehole with a prior art ceramic transducer (FIG. 5B) and the compositetransducer (FIG. 5A) in heavy water-based mud (18.2 ppg) is informative.The amplitude image 505 and the traveltime image 507 recorded with theceramic transducer show basically no reflected signal. In contrast, theamplitude image 501 and the traveltime image 503 with the compositetransducer shows a near vertical feature 521 indicative of boreholeellipticity. Based on experience with the composite transducer, it isestimated that the signal to noise ratio (SNR) can be increased by about20 dB by using a composite transducer. An explanation of the reductionin ringdown is that reverberations are dampened by the polymer matrixportion of the transducer

Another hardware solution is to alter the distance between thetransducer and the window. Changes in the reverberatory signal resultingfrom different distances have been discussed in Steinsiek but specificexamples of improvements in image quality were not discussed. Based onexperience with the composite transducer at different distances from thewindow, it is estimated that the SNR can be improved by about 8 dB.

Turning now to FIGS. 6A-6C, a processing method to improve the SNR isdiscussed. FIG. 6A shows an exemplary signal 601 that includes an echodistorted by ringdown. In this example, the echo is centered around 100on the x-axis. FIG. 6B shows an estimate 603 of the ringdown signalobtained by fitting a decaying sinusoid to the signal 601. In oneexample, the fitting function is of the form:

e ^(−ζt) sin(ωt−φ)

where the fitting parameters are the decay constant ζ, frequency ω, andphase φ. In one embodiment of the disclosure, a constrained fit is doneso that the curve 603 passes through the first peak 604 of the curve601. Subtracting the decaying sinusoidal fit 603 from the recordedsignal 601 gives the estimated echo signal 605. This process of fittingand subtracting (FAS) is repeated for each of the transducer excitationsat the plurality of transducer orientations and processed to giveamplitude and traveltime images.

Another method of processing the data is discussed with reference toFIGS. 7A-H. Four exemplary signals 701, 703, 705, 707 are shown in FIGS.7A-D. The signals include ringdown and an echo signal at time 90, 120,150 and 180 respectively. The average (“stack”) of the produced signals701, 703, 705, 707 is determined (not shown) and subtracted from theindividual signals 701, 703, 705, 707 to give the signals 701′, 703′,705′, 707′ respectively, shown in FIGS. 7E-H. As can be seen, thesubtracted signals agree very well with the echo times of 90, 120, 150and 180.

FIG. 8A shows the results of processing a signal including a ringdownusing the ABS method. FIG. 8B shows the result of processing using SAS(Stack and Subtract) and ABS. FIG. 8C shows the result of FAS (Fit andSubtract), SAS and ABS.

Another method of processing the data uses echo peak detection methodsdiscussed in U.S. patent application Ser. No. 12/268,141 of Zhao, havingthe same assignee as the present disclosure and the contents of whichare incorporated herein by reference. The method of Zhao is discussedwith reference to FIGS. 9A-9C. Shown in FIG. 9A is a ringdown signal 903and an echo 901. The echo can be clearly distinguished from theringdown. In FIG. 9B, the echo 911 is barely visible relative to theringdown 913 and in FIG. 9C, the echo 921 is swamped by the ringdown923.

The method used in the present disclosure is to use the HilbertTransform to estimate the envelope of the recorded signal and thenidentify arrival times as peaks in the envelope that are indicative ofthe echo signal. As discussed in Zhao, the Hilbert transform isimplemented in conjunction with a Cauchy wavelet (a modulated Gaussianfilter) as a bandpass filter. For the particular transducer used, awavelet is defined having 5-6 cycles of a center frequency of 300 kHz, asampling interval of 4 MHz. Due to hardware constraints, the wavelet istruncated, and a Hanning weighting is used to reduce the Gibbsphenomenon. The resulting wavelet pair for applying the Hilberttransform is shown in FIG. 10 by the in-phase 1003 and quadrature 1001wavelets.

FIG. 11A shows an exemplary signal 1101 completely dominated by theringdown. FIG. 11B shows the envelope 1103 obtained by applying thewavelets 1001, 1003. The perturbation 1105 is associated with the echosignal. Those versed in the art and having benefit of the presentdisclosure would recognize that peak finding techniques would not alwaysbe able to detect the echo. Accordingly, in one embodiment of thedisclosure, the first and second moments are removed from the envelopecurve using a Laplace Operator. The Laplace operator may be denoted by:

$\begin{matrix}{\nabla^{2}{= {\frac{d^{2}}{d\; t^{2}}.}}} & (1)\end{matrix}$

This filter is very sensitive to high frequency noise, so that a lowpass filtering may be applied prior to the Laplace operator. In oneembodiment of the disclosure, a Gaussian filter is used, so that thecombination of the Gaussian-Laplace operator may be denoted by:

$\begin{matrix}{{\nabla^{2}{\cdot {g(t)}}} = {\frac{d^{2}}{d\; t^{2}}{^{- {(\frac{t}{\tau})}^{2}}.}}} & (2)\end{matrix}$

FIG. 12 shows the results of applying the Gaussian Laplace operator tothe envelope 1103. The echo signal 1201 can be clearly seen, separatefrom the ringdown signal. For the purposes of display of the boreholeimage, the amplitude of the echo is defined by the difference betweenthe peak and the average of the adjacent troughs. The time is defined bythe time of the peak.

Another embodiment of the disclosure uses a notch filtering operation toremove the effect of the reverberation. The geometry that causes thereverberation is shown in FIG. 13. The transducer 1303 is positioned ina circular borehole 1300 of radius R and is offset by a distance E fromthe center of the borehole. At an angle φ of the transducer, the point1313 at a distance r from the transducer defines the two-way traveltimefor the reflected transducer signal. The direction of rotation of theacoustic beam is indicated by 1311.

When E≠0 the radial length of the ultrasound beam r changes as afunction of θ. The change in r between each azimuth location isexpressed as dr/dθ. If tr is the time between each azimuth location,then the change in r between each azimuth location is given by dr/dtr.dr is determined by the instruments radial distance measurement, and dtris determined by the instruments transmit timing. The stationary noiseand the moving ultrasound echo constructively interfere to create peaksin the detected line scan data every wavelength (λ). Due to thepulse-echo technique, for every half-wavelength (λ/2) change in r, thetwo-way distance changes a full wavelength (λ).

Therefore, the period of interference Pi=((λ/2)/dr)·dtrThis gives the instantaneous frequency of the interference between theringdown signal and the echo as:

fi=(2·dr/λ)/dtr=(2/λ)·dr/dtr=(2/λ·dtr)·dr

An adjustable notch filter tuned to this frequency can be used to filterout the interference without filtering out the signal variation due tothe formation features.

The curve 1401 in FIG. 14 a gives the peak value of the reflected signalas the transducer rotates through 360°. FIG. 14 b shows the modeledreflected amplitude over a portion of the borehole of uniform radius.Using known methods, the instantaneous frequency of the reflected signalcan be estimated. The instantaneous frequency is the result ofinterference between the generated signal (having a known basefrequency) and a modulating signal caused by the interference, i.e., itwill have components at the sum snd difference of the base frequency andthe modulating frequency. The modulating frequency recovered isindicated by 1403. A corresponding notch filter is applied to therecorded signal as a function of azimuth to get rid of thereverberation.

FIG. 15A shows an exemplary amplitude image recorded downhole anddisplayed without any downhole processing. FIG. 15B shows the results ofusing SAS processing. Significant improvement is noted in the resolutionof the bedding in the regions indicated by 1501 and 1503. In fact, theestimated dip direction is opposite to that which would be inferred fromthe unprocessed section. The stationary noise of the woodgrain effecthas been greatly attenuated. To the extent that the underlying formationbedding is visible in the image, it can be said that the stationarynoise due to interference between the ringdown signal and the echo hasbeen substantially eliminated.

The processing of the data may be done by a downhole processor and/or asurface processor to give corrected measurements substantially in realtime. Implicit in the control and processing of the data is the use of acomputer program on a suitable machine readable medium that enables theprocessor to perform the control and processing. The machine readablemedium may include ROMs, EPROMs, EEPROMs, Flash Memories and Opticaldisks. Such media may also be used to store results of the processingdiscussed above.

What is claimed is:
 1. An apparatus configured to evaluate an earthformation from acoustic signals propagated into a borehole, theapparatus comprising: a transducer configured to produce a signalincluding a ringdown signal and an echo from a wall of the borehole froma plurality of orientations; and at least one processor configured toadjust the signal produced for each of the plurality of orientationsusing an estimation of the ringdown signal to increase a signal-to-noiseratio associated with the signal and use the adjusted signal to providea two-dimensional image of the earth formation such that thetwo-dimensional image is substantially free of a stationary noiseresulting from an interference between the ringdown signal and the echo.2. The apparatus of claim 1 wherein the at least one processor isfurther configured to provide the two-dimensional image by: (i) fittinga decaying sinusoid to the produced signal for each of the plurality oforientations, (ii) subtracting the decaying sinusoid from the producedsignal for each of the plurality of orientations to give a plurality ofprocessed signals, and (iii) using the plurality of processed signals toprovide the image.
 3. The apparatus of claim 1 wherein the at least oneprocessor is further configured to provide the two-dimensional image by:(i) determining an average of the produced signals for the plurality oforientations, (ii) subtracting the average of the produced signals fromthe produced signal for each of the plurality of orientations to give aplurality of processed signals, and (iii) using the plurality ofprocessed signals to provide the two-dimensional image.
 4. The apparatusof claim 1 wherein the at least one processor is further configured toprovide the two-dimensional image by: (i) bandpassing the producedsignal for each of the plurality of orientations using a modulatedGaussian filter and providing a bandpassed signal for each of theplurality of orientations; (ii) estimating an envelope of the bandpassedsignal for each of the plurality of orientations; (iii) estimating fromthe envelope of the bandpassed signal for each of the plurality oforientations an arrival time of the echo for each of the plurality oforientations and an associated amplitude for each of the plurality oforientations; and (iv) producing the image using at least one of: (i)the estimated arrival time for each of the plurality of orientations,and (ii) the estimated associated amplitude for each of the plurality oforientations.
 5. The apparatus of claim 1 wherein the at least oneprocessor is further configured to provide the two-dimensional image by:(i) estimating a frequency of an interference between the ringdownsignal and the echo as a function of azimuths, and (ii) performing afiltering operation using a notch filter derived using the estimatedfrequency of the interference between ringdown signal and the echosignal.
 6. The apparatus of claim 1 wherein the transducer is part of alogging string conveyed into the borehole on a wireline.
 7. A method ofevaluating an earth formation, the method comprising: conveying atransducer into a borehole; using the transducer to produce a signalincluding a ringdown signal and an echo from a wall of the borehole froma plurality of orientations; and adjusting the signal produced for eachof the plurality of orientations using an estimation of the ringdownsignal to increase a signal-to-noise ratio associated with the signaland using the adjusted signal to provide a two-dimensional image of theearth formation such that the two-dimensional image is substantiallyfree of a stationary noise resulting from interference between theringdown signal and the echo.
 8. The method of claim 7 wherein adjustingthe signal produced for each of the plurality of orientations furthercomprises: (i) fitting a decaying sinusoid to the produced signal foreach of the plurality of orientations, (ii) subtracting the decayingsinusoid from the produced signal for each of the plurality oforientations to give a plurality of processed signals, and (iii) usingthe plurality of processed signals to provide the two-dimensional image.9. The method of claim 7 wherein adjusting the signal produced for eachof the plurality of orientations further comprises: (i) determining anaverage of the produced signals for the plurality of orientations, (ii)subtracting the average of the produced signals from each producedsignal for each of the plurality of orientations to give a plurality ofprocessed signals, and (iii) using the plurality of processed signals toprovide the two-dimensional image.
 10. The method of claim 7 whereinadjusting the signal produced for each of the plurality of orientationsfurther comprises: (i) bandpassing the produced signal for each of theplurality of orientations using a modulated Gaussian filter andproviding a bandpassed signal for each of the plurality of orientations;(ii) estimating an envelope of the bandpassed signal for each of theplurality of orientations; (iii) estimating from the envelope of thebandpassed signal for each of the plurality of orientations an arrivaltime of the echo for each of the orientations and an associatedamplitude for each of the plurality of orientations; and (iv) producingthe image using at least one of: (i) the estimated arrival time for eachof the plurality of orientations, and (ii) the estimated associatedamplitude for each of the plurality of orientations.
 11. The method ofclaim 7 wherein adjusting the signal produced for each of the pluralityof orientations further comprises: (i) estimating a frequency of aninterference between the ringdown signal and the echo as a function ofazimuths, and (ii) performing a filtering operation using a notch filterderived using the estimated frequency of the interference betweenringdown signal and the echo signal.
 12. The method of claim 7 furthercomprising conveying the transducer on a logging string into theborehole.
 13. A non-transitory computer-readable medium product havingstored thereon instructions that when executed by at least oneprocessor, cause the at least one processor to perform a method, themethod comprising: adjusting a signal produced by a transducer in aborehole, the signal including a ringdown signal and an echo from a wallof the borehole from a plurality of orientations, the adjusting using anestimation of the ringdown signal to increase a signal-to-noise ratioassociated with the signal, and using the adjusted signal to provide atwo-dimensional image of the earth formation such that thetwo-dimensional image is substantially free of a stationary noiseresulting from interference between the ringdown signal and the echo.14. The non-transitory computer-readable medium product of claim 13wherein adjusting the signal produced further comprises: (i) fitting adecaying sinusoid to the produced signal for each of the plurality oforientations, (ii) subtracting the decaying sinusoid from the producedsignal for each of the plurality of orientations to give a plurality ofprocessed signals, and (iii) using the plurality of processed signals toprovide the two-dimensional image.
 15. The non-transitorycomputer-readable medium product of claim 13 wherein adjusting thesignal produced further comprises: (i) determining an average of theproduced signals for the plurality of orientations, (ii) subtracting theaverage of the produced signals from each produced signal for each ofthe plurality of orientations to give a plurality of processed signals,and (iii) using the plurality of processed signals to provide thetwo-dimensional image.
 16. The non-transitory computer-readable mediumproduct of claim 13 wherein adjusting the signal produced furthercomprises: (i) bandpassing the produced signal for each of the pluralityof orientations using a modulated Gaussian filter and providing abandpassed signal for each of the plurality of orientations; (ii)estimating an envelope of the bandpassed signal for each of theplurality of orientations; (iii) estimating from the envelope of thebandpassed signal for each of the plurality of orientations an arrivaltime of the echo for each of the orientations and an associatedamplitude for each of the plurality of orientations; and (iv) producingthe image using at least one of: (i) the estimated arrival time for eachof the plurality of orientations, and (ii) the estimated associatedamplitude for each of the plurality of orientations.
 17. Thenon-transitory computer-readable medium product of claim 13 whereinadjusting the signal produced further comprises: (i) estimating afrequency of an interference between the ringdown signal and the echo asa function of azimuths, and (ii) performing a filtering operation usinga notch filter derived using the estimated frequency of the interferencebetween ringdown signal and the echo signal.
 18. The non-transitorycomputer-readable medium product of claim 13 further comprising at leastone of: (i) a ROM, (ii) an EPROM, (iii) an EEPROM, (iv) a flash memory,and (v) an optical disk.